{"slug": "certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal", "title": "Certification from Examples is Hard for Circuits and Transformers under Minimal Overparametrization", "summary": "New research shows that exact certification of neural networks—determining the smallest set of labeled examples needed to guarantee a learned hypothesis matches the target—becomes exponentially hard under minimal overparametrization. For threshold circuits with at least two layers, adding a single extra gate forces certificate sizes to grow exponentially with input dimension, and log-precision Transformers with constant architectural overhead face analogous hardness. The findings reveal that even approximate certification requiring only polynomially many mistakes still demands exponentially large certificates, while imperfect models can evade detection by large uniformly sampled certificate candidates.", "body_md": "arXiv:2605.22964v1 Announce Type: new\nAbstract: As state-of-the-art neural networks are deployed on reasoning and algorithmic tasks, exactness guarantees become increasingly important. However, high average-case accuracy can still mask inconsistent behaviors. This motivates exact certification, which asks for the smallest set of labeled examples needed to certify that a learned hypothesis equals the target. We show that while some hypotheses are easy to certify, even minimal overparametrization can make certification exponentially hard across several hypothesis classes. For threshold circuits of depth $\\ge 2$, adding a single extra gate can force certificate sizes exponential in the input dimension. We show an analogous hardness result for log-precision Transformers with only constant architectural overhead. We also characterize approximate certification, showing that allowing only polynomially many mistakes still requires exponentially large certificates, whereas constant relative-error guarantees can hide exponentially many mistakes. Empirically, we study certification for constructed circuits and trained Transformers for recognizing binary addition. While the constructed circuits instantiate the exponential barrier for certification, the trained Transformer analysis shows that imperfect models can evade detection by large uniformly sampled certificate candidates.", "url": "https://wpnews.pro/news/certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal", "canonical_source": "https://arxiv.org/abs/2605.22964", "published_at": "2026-05-25 04:00:00+00:00", "updated_at": "2026-05-25 15:14:23.758152+00:00", "lang": "en", "topics": ["machine-learning", "neural-networks", "ai-research", "ai-safety", "artificial-intelligence"], "entities": ["arXiv"], "alternates": {"html": "https://wpnews.pro/news/certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal", "markdown": "https://wpnews.pro/news/certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal.md", "text": "https://wpnews.pro/news/certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal.txt", "jsonld": "https://wpnews.pro/news/certification-from-examples-is-hard-for-circuits-and-transformers-under-minimal.jsonld"}}